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Hindawi Publishing CorporationJournal of Quality and Reliability EngineeringVolume 2013 Article ID 302469 7 pageshttpdxdoiorg1011552013302469
Research ArticleAcceptance Sampling Plans from Life Tests Based onPercentiles of Half Normal Distribution
B Srinivasa Rao1 Ch Srinivasa Kumar2 and K Rosaiah3
1 Department of Mathematics amp Humanities RVR amp JC College of Engineering Guntur Aandhra Pardesh 522 019 India2Department of Mathematics KLUniversity Guntur Aandhra Pardesh 522 502 India3 Department of Statistics Acharya Nagarjuna University Guntur Aandhra Pardesh 522 010 India
Correspondence should be addressed to B Srinivasa Rao boyapatisrinuyahoocom
Received 15 April 2013 Revised 9 October 2013 Accepted 11 October 2013
Academic Editor Christian Kirchsteiger
Copyright copy 2013 B Srinivasa Rao et alThis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The design of acceptance sampling plans is developed under truncated life testing based on the percentiles of half normaldistribution The minimum sample size necessary to ensure the specified life percentile is obtained under a given consumerrsquos riskThe operating characteristic values of the sampling plans as well as the producerrsquos risk are presented The results are illustrated byexamples
1 Introduction
Acceptance sampling is concerned with inspection and deci-sion making regarding lots of products and constitutes oneof the oldest techniques in quality control If the lifetime ofthe product represents the quality characteristics of interestthe acceptance sampling is as follows a company receivesa shipment of product from a vendor This product isoften a component or raw material used in the companyrsquosmanufacturing process A sample is taken from the lot andthe relevant quality characteristic of the units in the sampleis inspected On the basis of the information in the samplea decision is made regarding lot disposition Traditionallywhen the life test indicates that the mean life of productsexceeds the specified one the lot of products is acceptedotherwise it is rejected Accepted lots are put into productionwhile rejected lots may be returned to the vendor or maybe subjected to some other lot disposition actions For thepurpose of reducing the test time and cost a truncated lifetest may be conducted to determine the sample size to ensurea certainmean life of products when the life test is terminatedat a time 119905
0 and the number of failures does not exceed a given
acceptance number 119888A common practice in life testing is to terminate the
life test by a predetermined time 1199050and note the number of
failures One of the objectives of these experiments is to set alower confidence limit on the mean life It is then to establish
a specified mean life with a given probability of at least 119901lowastwhich provides protection to consumers The test may beterminated before the time is reached or when the numberof failures exceeds the acceptance number 119888 in which case thedecision is to reject the lot
Studies regarding truncated life tests can be found inEpstein [1] Sobel and Tischendrof [2] Goode and Kao[3] Gupta and Groll [4] Gupta [5] Fertig and Mann [6]Kantam andRosaiah [7] Baklizi [8]Wu andTsai [9] Rosaiahand Kantam [10] Rosaiah et al [11] Tsai and Wu [12]Balakrishnan et al [13] Srinivasa Rao et al [14] SrinivasaRao et al [15] Aslam et al [16] and Srinivasa Rao et al[17] All these authors designed acceptance sampling plansbased on the mean life time under a truncated life testIn contrast Lio et al [18] considered acceptance samplingplans for percentiles using truncated life tests and assumingBirnbaum-Saunders distribution Srinivasa Rao and Kantam[19] developed similar plans for the percentiles of log-logisticdistribution
Normal distribution is the most preferred distribution instatistical studies But for life test models it is not suitabledistribution because of its domain [minusinfin +infin] Half normaldistribution is an increasing failure rate (IFR) model whichis most useful in reliability studies Because of this IFRnature with domain [0 +infin] we are motivated to study thisdistribution
2 Journal of Quality and Reliability Engineering
In this paper acceptance sampling plans are developed forpercentiles of half normal distribution life test and are givenin Section 2 Operating characteristic (OC) and producerrsquosrisk are given in Section 3 Examples based on real fatiguelife data sets are provided for an illustration in Section 4 Thepaper is closed with summary and conclusions in Section 5
2 Acceptance Sampling Plans
The probability density function (pdf) of a half normaldistribution is given by
119901 (119909) =2120579
120587119890minus11990921205792120587 119909 ge 0 (1)
Its cumulative distribution function (cdf) is
119875 (119909) = erf ( 120579119909radic120587) 119909 ge 0 (2)
where erf is an error function and 120579 is a parameter Assumethat the life time of a product follows half normal distributionwith 120590 as scale parameter Its cumulative distribution func-tion 119865(sdot) is given by
where 1205750= 119905119905119902 Here erfminus1 is simulated by
erfminus1 (119911) = 12
radic120587[119911 +120587
121199113+71205872
4801199115
+1271205873
403201199117+4369120587
4
58060801199119sdot sdot sdot ]
(6)
A common practice in life testing is to terminate the lifetest by a predetermined time 119905 to require the probability ofrejecting a bad lot to be at least 119901lowast and to have the maximumpermissible number of bad items to accept the lot equalto 119888 The acceptance sampling plan for percentiles under atruncated life test is to set up the minimum sample size 119899 fora given acceptance number 119888 such that the consumerrsquos riskthe probability of accepting a bad lot does not exceed 1 minus119901lowastA bad lot means that the true 100th percentile 119905
119902 is below a
specified percentile 1199050119902Thus the probability119901lowast is a confidence
level in the sense that the chance of rejecting a bad lot with
119905119902lt 1199050
119902is at least equal to 119901lowast Therefore for given 119901lowast the
proposed acceptance sampling plan can be characterized bythe triplet (119899 119888 120575
0) = (119899 119888 119905119905
0
119902) where 120575 = 1199051199050
119902
We consider large sized lots so that the binomial distri-bution can be applied The problem is to determine for givenvalues of 119901lowast 1199050
119902 and 119888 the smallest positive integer 119899 required
to assert that 119905119902gt 1199050
119902must satisfy the relation
119888
sum
119894=0
119901119894
0(1 minus 119901
0)119899minus119894le 1 minus 119901
lowast (7)
where 1199010= 119865119905(1205750) is the probability of a failure during the
time 119905 = 1205751199050119902for the specified 100qth percentile of lifetime 1199050
119902
The value 1199010depends only on 120575
0= 1199051199050
119902 Since 120597119865
119905(120575)120597120575 gt 0
it is an increasing function of 120575 Accordingly we have
The smallest sample size 119899 satisfying (9) can be obtained forany given 119902 1199051199050
119902 or 119901lowast In contrast only the input values
119905120590 119901lowast are needed to calculate the smallest sample size 119899 Inparticular if we take 119902 = 050 the value of 119899 would becomeldquothe smallest sample size required to test that the populationmedian life exceeds a given specified valuerdquo Half normaldistribution is a skewed distribution for our present skewedpopulation the median is a more appropriate average fordecision making about the quality of the life than populationmean Thus we may conclude that population median basedsampling plans of half normal distribution model are moreeconomical than those based on population mean withrespect to sample size To save space only the results of smallsample sizes for 119902 = 050 119901lowast = 075 090 095 099 119888 =0(1)10 120575 = 01 02 04 06 08 10 15 20 25 30 are dis-played in Table 1
3 Operating Characteristic of the SamplingPlan and Producerrsquos Risk
The operating characteristic (OC) function 119871 of the samplingplan (119899 119888 1199051199050
119902) is the probability of accepting a lot as a
function of 119901 = 119865119905(120575) with 120575
0= 119905119905119902 It is given as
119871 (119901) =
119888
sum
119894=0
(119899
119894) 119901119894(1 minus 119901)
119899minus119894 (10)
Therefore we have
119901 = 119865119905(120575) = 119865(
119905
1199050
119902
1
119889119902
) (11)
where 119889119902= 1199051199021199050
119902 Using (10) the OC values can be obtained
for any sampling plan (119899 119888 119905119905005) (Table 5) To save space we
Journal of Quality and Reliability Engineering 3
Table 1 Minimum sample sizes necessary to assert the median lifeof a product
present the OC values for sampling plans with 119902 = 050 119901lowast =075 090 095 099 120575 = 01020406081015202530119905051199050
05= 10 15 20 25 30 35 40 45 50 for 119888 = 1 and
119888 = 5 which are given in Table 2 and Table 3 respectively
The producerrsquos risk is defined as the probability ofrejecting the lot when 119905
119902gt 1199050
119902 For a given value of the
producerrsquos risk say 120574 we are interested in knowing the valueof 119889119902to ensure that the producerrsquos risk is less than or equal
to 120574 if a sampling plan (119899 119888 119905119905119902) is developed at a specified
confidence level119901lowastThus one needs to find the smallest value119889119902according to (10) as
119888
sum
119894=0
(119899
119894) 119901119894(1 minus 119901)
119899minus119894 (12)
where
119901 = 119865(119905
1199050
119902
1
119889119902
) (13)
To save space based on the sampling plans (119899 119888 1199051199050119902) given
in Table 1 the minimum ratios of 11988905
for the acceptability of alot at the producerrsquos risk of 120574 = 005 are presented in Table 4
4 Illustrative Examples
In this section we consider two examples with real data setsto illustrate the proposed acceptance sampling plans
41 Example 1 The first data refers to software reliabilitypresented by Wood [20] The data set was reported in hoursas 519 968 1430 1893 2490 3058 3625 4422 and 5218 Theconfidence level is assumed for this acceptance sampling planonly if the lifetimes are from half normal distribution withshape parameter 120590 So in order to apply this example for ourtables we have to confirm the goodness of fit of half normaldistribution to the data in the example This is done throughthe well-known QQ-plot method and the value we get is 119877 =09287 Hence we conclude that the half normal distributionprovides a reasonable goodness of fit for the data
Suppose that the experimenter would like to establish theunknown 50th percentile life time for the softwarementionedabove to be at least 300 h and the life test would be ended at600 h which should have led to the ratio 1199051199050
05= 20 Thus
with 119888 = 1 and 119901lowast = 075 the experimenter may take fromTable 1 the sample size 119899 which must be at least 3 thus thesampling plan to the data is given by
(119899 119888119905
1199050
05
) = (3 1 20) (14)
Since there is one itemwith a failure time less than or equal to600 h in the given sample of 9 observations the lot is acceptedas the result indicates that the 50th percentile life time 119905
05is
at least 300 h with a confidence level of 119901lowast = 075The OC values for acceptance sampling plan in (14) and
confidence 119901lowast = 075 for a half normal distribution may betaken from Table 2
The producerrsquos risk is almost equal to 03683 (= 1 minus06317) when the true percentile is greater than or equal to20 times the specified 50th percentile
From Table 4 the experimenter could get the values of11988905
for different choices of 119888 and 119905119905005
in order to assert thatthe producerrsquos risk is less than 005 In this example the valueof11988905
should be 79120 for 119888 = 1 119905051199050
05= 20 and119901lowast = 075
This means that the product can have a life of 79120 timesthe required 50th percentile life time in order that under theabove acceptance sampling plan the product is accepted withprobability at least 075
42 Example 2 The second data refers to the data obtainedfrom Aarset [21] It represents the lifetimes of 50 devices inhours The data set was reported in hours as 01 02 1 1 1 11 2 3 6 7 11 12 18 18 18 18 18 21 32 36 40 45 46 47 5055 60 63 63 67 67 67 67 72 75 79 82 82 83 84 84 84 8585 85 85 85 86 and 86 On the similar lines of Example 1for this data the goodness of fit for 50 observations showedthat 119877 = 09175
Suppose that wewould like to establish the unknown 50thpercentile life time for the devices to be at least 10 hours andthe life test would be ended at 15 hrs which should have ledto the ratio of 119905
051199050
05= 15 Thus with 119888 = 5 119901lowast = 099
the experimenter may take from Table 1 the sample size 119899whichmust be at least 15Thus the sampling plan to the givendata is given by (119899 119888 1199051199050
05) = (15 5 15) Since there are 13
items with a failure time less than or equal to 15 hours inthe given sample of 50 observations the lot is accepted as the
result indicates that the 50th percentile lifetime 11990505
is at least10 hours with a confidence level of 119901lowast = 099
5 Summary and Conclusions
This paper provides the minimum sample size required todecide upon acceptingrejecting a lot based on its specified50th percentile (median) when the data follows half normaldistribution Assuming that the size of the given sample
6 Journal of Quality and Reliability Engineering
Table 4 Minimum ratio for the acceptability of a lot with producerrsquos risk 005
is minimum we can use the tables in this paper to pickup acceptance number 119888 and the upper bound in order toacceptreject the lot for a given probability of acceptance say119901lowast This paper also provides the sensitivity of the sampling
plans in terms of theOCWe advise the industrial practitionerand the experimenter to adopt this plan in order to save thecost and time of the experiment This plan can be furtherstudied for many other distributions and various quality andreliability characteristics as future research
Acknowledgments
The authors thank the editor and the reviewers for theirhelpful suggestions comments and encouragement whichhelped in improving the final version of the paper
References
[1] B Epstein ldquoTruncated life tests in the exponential caserdquo Annalsof Mathematical Statistics vol 25 pp 555ndash564 1954
[2] M Sobel and J A Tischendrof ldquoAcceptance samplingwith skewlife test objectiverdquo inProceedings of the 5thNational SyposiumonReliability and Qualilty Control pp 108ndash118 Philadelphia PaUSA 1959
[3] H P Goode and J H K Kao ldquoSampling plans based on theWeibull distributionrdquo in Proceedings of the 7th National Sypo-sium on Reliability and Qualilty Control pp 24ndash40 Philadel-phia Pa USA 1961
[4] S S Gupta and P A Groll ldquoGamma distribution in acceptancesampling based on life testsrdquo Journal of the American StatisticalAssociation vol 56 pp 942ndash970 1961
[5] S S Gupta ldquoLife test sampling plans for normal and lognormaldistributionrdquo Technometrics vol 4 pp 151ndash175 1962
[6] F W Fertig and N R Mann ldquoLife-test sampling plans for two-parameter Weibull populationsrdquo Technometrics vol 22 pp165ndash177 1980
[7] R R L Kantam and K Rosaiah ldquoHalf Logistic distribution inacceptance sampling based on life testsrdquo IAPQR Transactionsvol 23 no 2 pp 117ndash125 1998
[8] A Baklizi ldquoAcceptance sampling based on truncated life testsin the pareto distribution of the second kindrdquo Advances andApplications in Statistics vol 3 pp 33ndash48 2003
[9] C-J Wu and T-R Tsai ldquoAcceptance sampling plans forbirnbaum-saunders distribution under truncated life testsrdquoInternational Journal of Reliability Quality and Safety Engineer-ing vol 12 no 6 pp 507ndash519 2005
[10] K Rosaiah and R R L Kantam ldquoAcceptance sampling plansbased on inverse Rayleigh distributionrdquo Economic QualityControl vol 20 no 2 pp 277ndash286 2005
[11] K Rosaiah R R L Kantam and Ch Santosh Kumar ldquoReli-ability test plans for exponnetiated log-logistic distributionrdquoEconomic Quality Control vol 21 pp 165ndash175 2006
[12] T-R Tsai and S-J Wu ldquoAcceptance sampling based on trun-cated life tests for generalized Rayleigh distributionrdquo Journal ofApplied Statistics vol 33 no 6 pp 595ndash600 2006
[13] N Balakrishnan V Leiva and J Lopez ldquoAcceptance sam-pling plans from truncated life tests based on the generalizedBirnbaum-Saunders distributionrdquo Communications in Statis-tics Simulation and Computation vol 36 no 3 pp 643ndash6562007
[14] G Srinivasa Rao M E Ghitany and R R L KantamldquoReliability test plans for Marshall-Olkin extended exponentialdistributionrdquo Applied Mathematical Sciences vol 3 no 53-56pp 2745ndash2755 2009
[15] G Srinivasa Rao M E Ghitany and R R L Kantam ldquoAneconomic reliability test plan for Marshall-Olkin extendedexponential distributionrdquoAppliedMathematical Sciences vol 5no 1ndash4 pp 103ndash112 2011
[16] M Aslam C-H Jun and M Ahmad ldquoA group samplingplan based on truncated life test for gamma distributed itemsrdquoPakistan Journal of Statistics vol 25 no 3 pp 333ndash340 2009
[17] G Srinivasa Rao M E Ghitany and R R L KantamldquoReliability test plans for Marshall-Olkin extended exponentialdistributionrdquo Applied Mathematical Sciences vol 3 no 53-56pp 2745ndash2755 2009
[18] Y L Lio T-R Tsai and S-J Wu ldquoAcceptance sampling plansfrom truncated life tests based on the birnbaum-saunders distri-bution for percentilesrdquoCommunications in Statistics Simulationand Computation vol 39 no 1 pp 119ndash136 2010
[19] G Srinivasa Rao and R R L Kantam ldquoAcceptance samplingplans from truncated life tests based on log-logistic distributionfor percentilesrdquoEconomicQuality Control vol 25 no 2 pp 153ndash167 2010
In this paper acceptance sampling plans are developed forpercentiles of half normal distribution life test and are givenin Section 2 Operating characteristic (OC) and producerrsquosrisk are given in Section 3 Examples based on real fatiguelife data sets are provided for an illustration in Section 4 Thepaper is closed with summary and conclusions in Section 5
2 Acceptance Sampling Plans
The probability density function (pdf) of a half normaldistribution is given by
119901 (119909) =2120579
120587119890minus11990921205792120587 119909 ge 0 (1)
Its cumulative distribution function (cdf) is
119875 (119909) = erf ( 120579119909radic120587) 119909 ge 0 (2)
where erf is an error function and 120579 is a parameter Assumethat the life time of a product follows half normal distributionwith 120590 as scale parameter Its cumulative distribution func-tion 119865(sdot) is given by
where 1205750= 119905119905119902 Here erfminus1 is simulated by
erfminus1 (119911) = 12
radic120587[119911 +120587
121199113+71205872
4801199115
+1271205873
403201199117+4369120587
4
58060801199119sdot sdot sdot ]
(6)
A common practice in life testing is to terminate the lifetest by a predetermined time 119905 to require the probability ofrejecting a bad lot to be at least 119901lowast and to have the maximumpermissible number of bad items to accept the lot equalto 119888 The acceptance sampling plan for percentiles under atruncated life test is to set up the minimum sample size 119899 fora given acceptance number 119888 such that the consumerrsquos riskthe probability of accepting a bad lot does not exceed 1 minus119901lowastA bad lot means that the true 100th percentile 119905
119902 is below a
specified percentile 1199050119902Thus the probability119901lowast is a confidence
level in the sense that the chance of rejecting a bad lot with
119905119902lt 1199050
119902is at least equal to 119901lowast Therefore for given 119901lowast the
proposed acceptance sampling plan can be characterized bythe triplet (119899 119888 120575
0) = (119899 119888 119905119905
0
119902) where 120575 = 1199051199050
119902
We consider large sized lots so that the binomial distri-bution can be applied The problem is to determine for givenvalues of 119901lowast 1199050
119902 and 119888 the smallest positive integer 119899 required
to assert that 119905119902gt 1199050
119902must satisfy the relation
119888
sum
119894=0
119901119894
0(1 minus 119901
0)119899minus119894le 1 minus 119901
lowast (7)
where 1199010= 119865119905(1205750) is the probability of a failure during the
time 119905 = 1205751199050119902for the specified 100qth percentile of lifetime 1199050
119902
The value 1199010depends only on 120575
0= 1199051199050
119902 Since 120597119865
119905(120575)120597120575 gt 0
it is an increasing function of 120575 Accordingly we have
The smallest sample size 119899 satisfying (9) can be obtained forany given 119902 1199051199050
119902 or 119901lowast In contrast only the input values
119905120590 119901lowast are needed to calculate the smallest sample size 119899 Inparticular if we take 119902 = 050 the value of 119899 would becomeldquothe smallest sample size required to test that the populationmedian life exceeds a given specified valuerdquo Half normaldistribution is a skewed distribution for our present skewedpopulation the median is a more appropriate average fordecision making about the quality of the life than populationmean Thus we may conclude that population median basedsampling plans of half normal distribution model are moreeconomical than those based on population mean withrespect to sample size To save space only the results of smallsample sizes for 119902 = 050 119901lowast = 075 090 095 099 119888 =0(1)10 120575 = 01 02 04 06 08 10 15 20 25 30 are dis-played in Table 1
3 Operating Characteristic of the SamplingPlan and Producerrsquos Risk
The operating characteristic (OC) function 119871 of the samplingplan (119899 119888 1199051199050
119902) is the probability of accepting a lot as a
function of 119901 = 119865119905(120575) with 120575
0= 119905119905119902 It is given as
119871 (119901) =
119888
sum
119894=0
(119899
119894) 119901119894(1 minus 119901)
119899minus119894 (10)
Therefore we have
119901 = 119865119905(120575) = 119865(
119905
1199050
119902
1
119889119902
) (11)
where 119889119902= 1199051199021199050
119902 Using (10) the OC values can be obtained
for any sampling plan (119899 119888 119905119905005) (Table 5) To save space we
Journal of Quality and Reliability Engineering 3
Table 1 Minimum sample sizes necessary to assert the median lifeof a product
present the OC values for sampling plans with 119902 = 050 119901lowast =075 090 095 099 120575 = 01020406081015202530119905051199050
05= 10 15 20 25 30 35 40 45 50 for 119888 = 1 and
119888 = 5 which are given in Table 2 and Table 3 respectively
The producerrsquos risk is defined as the probability ofrejecting the lot when 119905
119902gt 1199050
119902 For a given value of the
producerrsquos risk say 120574 we are interested in knowing the valueof 119889119902to ensure that the producerrsquos risk is less than or equal
to 120574 if a sampling plan (119899 119888 119905119905119902) is developed at a specified
confidence level119901lowastThus one needs to find the smallest value119889119902according to (10) as
119888
sum
119894=0
(119899
119894) 119901119894(1 minus 119901)
119899minus119894 (12)
where
119901 = 119865(119905
1199050
119902
1
119889119902
) (13)
To save space based on the sampling plans (119899 119888 1199051199050119902) given
in Table 1 the minimum ratios of 11988905
for the acceptability of alot at the producerrsquos risk of 120574 = 005 are presented in Table 4
4 Illustrative Examples
In this section we consider two examples with real data setsto illustrate the proposed acceptance sampling plans
41 Example 1 The first data refers to software reliabilitypresented by Wood [20] The data set was reported in hoursas 519 968 1430 1893 2490 3058 3625 4422 and 5218 Theconfidence level is assumed for this acceptance sampling planonly if the lifetimes are from half normal distribution withshape parameter 120590 So in order to apply this example for ourtables we have to confirm the goodness of fit of half normaldistribution to the data in the example This is done throughthe well-known QQ-plot method and the value we get is 119877 =09287 Hence we conclude that the half normal distributionprovides a reasonable goodness of fit for the data
Suppose that the experimenter would like to establish theunknown 50th percentile life time for the softwarementionedabove to be at least 300 h and the life test would be ended at600 h which should have led to the ratio 1199051199050
05= 20 Thus
with 119888 = 1 and 119901lowast = 075 the experimenter may take fromTable 1 the sample size 119899 which must be at least 3 thus thesampling plan to the data is given by
(119899 119888119905
1199050
05
) = (3 1 20) (14)
Since there is one itemwith a failure time less than or equal to600 h in the given sample of 9 observations the lot is acceptedas the result indicates that the 50th percentile life time 119905
05is
at least 300 h with a confidence level of 119901lowast = 075The OC values for acceptance sampling plan in (14) and
confidence 119901lowast = 075 for a half normal distribution may betaken from Table 2
The producerrsquos risk is almost equal to 03683 (= 1 minus06317) when the true percentile is greater than or equal to20 times the specified 50th percentile
From Table 4 the experimenter could get the values of11988905
for different choices of 119888 and 119905119905005
in order to assert thatthe producerrsquos risk is less than 005 In this example the valueof11988905
should be 79120 for 119888 = 1 119905051199050
05= 20 and119901lowast = 075
This means that the product can have a life of 79120 timesthe required 50th percentile life time in order that under theabove acceptance sampling plan the product is accepted withprobability at least 075
42 Example 2 The second data refers to the data obtainedfrom Aarset [21] It represents the lifetimes of 50 devices inhours The data set was reported in hours as 01 02 1 1 1 11 2 3 6 7 11 12 18 18 18 18 18 21 32 36 40 45 46 47 5055 60 63 63 67 67 67 67 72 75 79 82 82 83 84 84 84 8585 85 85 85 86 and 86 On the similar lines of Example 1for this data the goodness of fit for 50 observations showedthat 119877 = 09175
Suppose that wewould like to establish the unknown 50thpercentile life time for the devices to be at least 10 hours andthe life test would be ended at 15 hrs which should have ledto the ratio of 119905
051199050
05= 15 Thus with 119888 = 5 119901lowast = 099
the experimenter may take from Table 1 the sample size 119899whichmust be at least 15Thus the sampling plan to the givendata is given by (119899 119888 1199051199050
05) = (15 5 15) Since there are 13
items with a failure time less than or equal to 15 hours inthe given sample of 50 observations the lot is accepted as the
result indicates that the 50th percentile lifetime 11990505
is at least10 hours with a confidence level of 119901lowast = 099
5 Summary and Conclusions
This paper provides the minimum sample size required todecide upon acceptingrejecting a lot based on its specified50th percentile (median) when the data follows half normaldistribution Assuming that the size of the given sample
6 Journal of Quality and Reliability Engineering
Table 4 Minimum ratio for the acceptability of a lot with producerrsquos risk 005
is minimum we can use the tables in this paper to pickup acceptance number 119888 and the upper bound in order toacceptreject the lot for a given probability of acceptance say119901lowast This paper also provides the sensitivity of the sampling
plans in terms of theOCWe advise the industrial practitionerand the experimenter to adopt this plan in order to save thecost and time of the experiment This plan can be furtherstudied for many other distributions and various quality andreliability characteristics as future research
Acknowledgments
The authors thank the editor and the reviewers for theirhelpful suggestions comments and encouragement whichhelped in improving the final version of the paper
References
[1] B Epstein ldquoTruncated life tests in the exponential caserdquo Annalsof Mathematical Statistics vol 25 pp 555ndash564 1954
[2] M Sobel and J A Tischendrof ldquoAcceptance samplingwith skewlife test objectiverdquo inProceedings of the 5thNational SyposiumonReliability and Qualilty Control pp 108ndash118 Philadelphia PaUSA 1959
[3] H P Goode and J H K Kao ldquoSampling plans based on theWeibull distributionrdquo in Proceedings of the 7th National Sypo-sium on Reliability and Qualilty Control pp 24ndash40 Philadel-phia Pa USA 1961
[4] S S Gupta and P A Groll ldquoGamma distribution in acceptancesampling based on life testsrdquo Journal of the American StatisticalAssociation vol 56 pp 942ndash970 1961
[5] S S Gupta ldquoLife test sampling plans for normal and lognormaldistributionrdquo Technometrics vol 4 pp 151ndash175 1962
[6] F W Fertig and N R Mann ldquoLife-test sampling plans for two-parameter Weibull populationsrdquo Technometrics vol 22 pp165ndash177 1980
[7] R R L Kantam and K Rosaiah ldquoHalf Logistic distribution inacceptance sampling based on life testsrdquo IAPQR Transactionsvol 23 no 2 pp 117ndash125 1998
[8] A Baklizi ldquoAcceptance sampling based on truncated life testsin the pareto distribution of the second kindrdquo Advances andApplications in Statistics vol 3 pp 33ndash48 2003
[9] C-J Wu and T-R Tsai ldquoAcceptance sampling plans forbirnbaum-saunders distribution under truncated life testsrdquoInternational Journal of Reliability Quality and Safety Engineer-ing vol 12 no 6 pp 507ndash519 2005
[10] K Rosaiah and R R L Kantam ldquoAcceptance sampling plansbased on inverse Rayleigh distributionrdquo Economic QualityControl vol 20 no 2 pp 277ndash286 2005
[11] K Rosaiah R R L Kantam and Ch Santosh Kumar ldquoReli-ability test plans for exponnetiated log-logistic distributionrdquoEconomic Quality Control vol 21 pp 165ndash175 2006
[12] T-R Tsai and S-J Wu ldquoAcceptance sampling based on trun-cated life tests for generalized Rayleigh distributionrdquo Journal ofApplied Statistics vol 33 no 6 pp 595ndash600 2006
[13] N Balakrishnan V Leiva and J Lopez ldquoAcceptance sam-pling plans from truncated life tests based on the generalizedBirnbaum-Saunders distributionrdquo Communications in Statis-tics Simulation and Computation vol 36 no 3 pp 643ndash6562007
[14] G Srinivasa Rao M E Ghitany and R R L KantamldquoReliability test plans for Marshall-Olkin extended exponentialdistributionrdquo Applied Mathematical Sciences vol 3 no 53-56pp 2745ndash2755 2009
[15] G Srinivasa Rao M E Ghitany and R R L Kantam ldquoAneconomic reliability test plan for Marshall-Olkin extendedexponential distributionrdquoAppliedMathematical Sciences vol 5no 1ndash4 pp 103ndash112 2011
[16] M Aslam C-H Jun and M Ahmad ldquoA group samplingplan based on truncated life test for gamma distributed itemsrdquoPakistan Journal of Statistics vol 25 no 3 pp 333ndash340 2009
[17] G Srinivasa Rao M E Ghitany and R R L KantamldquoReliability test plans for Marshall-Olkin extended exponentialdistributionrdquo Applied Mathematical Sciences vol 3 no 53-56pp 2745ndash2755 2009
[18] Y L Lio T-R Tsai and S-J Wu ldquoAcceptance sampling plansfrom truncated life tests based on the birnbaum-saunders distri-bution for percentilesrdquoCommunications in Statistics Simulationand Computation vol 39 no 1 pp 119ndash136 2010
[19] G Srinivasa Rao and R R L Kantam ldquoAcceptance samplingplans from truncated life tests based on log-logistic distributionfor percentilesrdquoEconomicQuality Control vol 25 no 2 pp 153ndash167 2010
present the OC values for sampling plans with 119902 = 050 119901lowast =075 090 095 099 120575 = 01020406081015202530119905051199050
05= 10 15 20 25 30 35 40 45 50 for 119888 = 1 and
119888 = 5 which are given in Table 2 and Table 3 respectively
The producerrsquos risk is defined as the probability ofrejecting the lot when 119905
119902gt 1199050
119902 For a given value of the
producerrsquos risk say 120574 we are interested in knowing the valueof 119889119902to ensure that the producerrsquos risk is less than or equal
to 120574 if a sampling plan (119899 119888 119905119905119902) is developed at a specified
confidence level119901lowastThus one needs to find the smallest value119889119902according to (10) as
119888
sum
119894=0
(119899
119894) 119901119894(1 minus 119901)
119899minus119894 (12)
where
119901 = 119865(119905
1199050
119902
1
119889119902
) (13)
To save space based on the sampling plans (119899 119888 1199051199050119902) given
in Table 1 the minimum ratios of 11988905
for the acceptability of alot at the producerrsquos risk of 120574 = 005 are presented in Table 4
4 Illustrative Examples
In this section we consider two examples with real data setsto illustrate the proposed acceptance sampling plans
41 Example 1 The first data refers to software reliabilitypresented by Wood [20] The data set was reported in hoursas 519 968 1430 1893 2490 3058 3625 4422 and 5218 Theconfidence level is assumed for this acceptance sampling planonly if the lifetimes are from half normal distribution withshape parameter 120590 So in order to apply this example for ourtables we have to confirm the goodness of fit of half normaldistribution to the data in the example This is done throughthe well-known QQ-plot method and the value we get is 119877 =09287 Hence we conclude that the half normal distributionprovides a reasonable goodness of fit for the data
Suppose that the experimenter would like to establish theunknown 50th percentile life time for the softwarementionedabove to be at least 300 h and the life test would be ended at600 h which should have led to the ratio 1199051199050
05= 20 Thus
with 119888 = 1 and 119901lowast = 075 the experimenter may take fromTable 1 the sample size 119899 which must be at least 3 thus thesampling plan to the data is given by
(119899 119888119905
1199050
05
) = (3 1 20) (14)
Since there is one itemwith a failure time less than or equal to600 h in the given sample of 9 observations the lot is acceptedas the result indicates that the 50th percentile life time 119905
05is
at least 300 h with a confidence level of 119901lowast = 075The OC values for acceptance sampling plan in (14) and
confidence 119901lowast = 075 for a half normal distribution may betaken from Table 2
The producerrsquos risk is almost equal to 03683 (= 1 minus06317) when the true percentile is greater than or equal to20 times the specified 50th percentile
From Table 4 the experimenter could get the values of11988905
for different choices of 119888 and 119905119905005
in order to assert thatthe producerrsquos risk is less than 005 In this example the valueof11988905
should be 79120 for 119888 = 1 119905051199050
05= 20 and119901lowast = 075
This means that the product can have a life of 79120 timesthe required 50th percentile life time in order that under theabove acceptance sampling plan the product is accepted withprobability at least 075
42 Example 2 The second data refers to the data obtainedfrom Aarset [21] It represents the lifetimes of 50 devices inhours The data set was reported in hours as 01 02 1 1 1 11 2 3 6 7 11 12 18 18 18 18 18 21 32 36 40 45 46 47 5055 60 63 63 67 67 67 67 72 75 79 82 82 83 84 84 84 8585 85 85 85 86 and 86 On the similar lines of Example 1for this data the goodness of fit for 50 observations showedthat 119877 = 09175
Suppose that wewould like to establish the unknown 50thpercentile life time for the devices to be at least 10 hours andthe life test would be ended at 15 hrs which should have ledto the ratio of 119905
051199050
05= 15 Thus with 119888 = 5 119901lowast = 099
the experimenter may take from Table 1 the sample size 119899whichmust be at least 15Thus the sampling plan to the givendata is given by (119899 119888 1199051199050
05) = (15 5 15) Since there are 13
items with a failure time less than or equal to 15 hours inthe given sample of 50 observations the lot is accepted as the
result indicates that the 50th percentile lifetime 11990505
is at least10 hours with a confidence level of 119901lowast = 099
5 Summary and Conclusions
This paper provides the minimum sample size required todecide upon acceptingrejecting a lot based on its specified50th percentile (median) when the data follows half normaldistribution Assuming that the size of the given sample
6 Journal of Quality and Reliability Engineering
Table 4 Minimum ratio for the acceptability of a lot with producerrsquos risk 005
is minimum we can use the tables in this paper to pickup acceptance number 119888 and the upper bound in order toacceptreject the lot for a given probability of acceptance say119901lowast This paper also provides the sensitivity of the sampling
plans in terms of theOCWe advise the industrial practitionerand the experimenter to adopt this plan in order to save thecost and time of the experiment This plan can be furtherstudied for many other distributions and various quality andreliability characteristics as future research
Acknowledgments
The authors thank the editor and the reviewers for theirhelpful suggestions comments and encouragement whichhelped in improving the final version of the paper
References
[1] B Epstein ldquoTruncated life tests in the exponential caserdquo Annalsof Mathematical Statistics vol 25 pp 555ndash564 1954
[2] M Sobel and J A Tischendrof ldquoAcceptance samplingwith skewlife test objectiverdquo inProceedings of the 5thNational SyposiumonReliability and Qualilty Control pp 108ndash118 Philadelphia PaUSA 1959
[3] H P Goode and J H K Kao ldquoSampling plans based on theWeibull distributionrdquo in Proceedings of the 7th National Sypo-sium on Reliability and Qualilty Control pp 24ndash40 Philadel-phia Pa USA 1961
[4] S S Gupta and P A Groll ldquoGamma distribution in acceptancesampling based on life testsrdquo Journal of the American StatisticalAssociation vol 56 pp 942ndash970 1961
[5] S S Gupta ldquoLife test sampling plans for normal and lognormaldistributionrdquo Technometrics vol 4 pp 151ndash175 1962
[6] F W Fertig and N R Mann ldquoLife-test sampling plans for two-parameter Weibull populationsrdquo Technometrics vol 22 pp165ndash177 1980
[7] R R L Kantam and K Rosaiah ldquoHalf Logistic distribution inacceptance sampling based on life testsrdquo IAPQR Transactionsvol 23 no 2 pp 117ndash125 1998
[8] A Baklizi ldquoAcceptance sampling based on truncated life testsin the pareto distribution of the second kindrdquo Advances andApplications in Statistics vol 3 pp 33ndash48 2003
[9] C-J Wu and T-R Tsai ldquoAcceptance sampling plans forbirnbaum-saunders distribution under truncated life testsrdquoInternational Journal of Reliability Quality and Safety Engineer-ing vol 12 no 6 pp 507ndash519 2005
[10] K Rosaiah and R R L Kantam ldquoAcceptance sampling plansbased on inverse Rayleigh distributionrdquo Economic QualityControl vol 20 no 2 pp 277ndash286 2005
[11] K Rosaiah R R L Kantam and Ch Santosh Kumar ldquoReli-ability test plans for exponnetiated log-logistic distributionrdquoEconomic Quality Control vol 21 pp 165ndash175 2006
[12] T-R Tsai and S-J Wu ldquoAcceptance sampling based on trun-cated life tests for generalized Rayleigh distributionrdquo Journal ofApplied Statistics vol 33 no 6 pp 595ndash600 2006
[13] N Balakrishnan V Leiva and J Lopez ldquoAcceptance sam-pling plans from truncated life tests based on the generalizedBirnbaum-Saunders distributionrdquo Communications in Statis-tics Simulation and Computation vol 36 no 3 pp 643ndash6562007
[14] G Srinivasa Rao M E Ghitany and R R L KantamldquoReliability test plans for Marshall-Olkin extended exponentialdistributionrdquo Applied Mathematical Sciences vol 3 no 53-56pp 2745ndash2755 2009
[15] G Srinivasa Rao M E Ghitany and R R L Kantam ldquoAneconomic reliability test plan for Marshall-Olkin extendedexponential distributionrdquoAppliedMathematical Sciences vol 5no 1ndash4 pp 103ndash112 2011
[16] M Aslam C-H Jun and M Ahmad ldquoA group samplingplan based on truncated life test for gamma distributed itemsrdquoPakistan Journal of Statistics vol 25 no 3 pp 333ndash340 2009
[17] G Srinivasa Rao M E Ghitany and R R L KantamldquoReliability test plans for Marshall-Olkin extended exponentialdistributionrdquo Applied Mathematical Sciences vol 3 no 53-56pp 2745ndash2755 2009
[18] Y L Lio T-R Tsai and S-J Wu ldquoAcceptance sampling plansfrom truncated life tests based on the birnbaum-saunders distri-bution for percentilesrdquoCommunications in Statistics Simulationand Computation vol 39 no 1 pp 119ndash136 2010
[19] G Srinivasa Rao and R R L Kantam ldquoAcceptance samplingplans from truncated life tests based on log-logistic distributionfor percentilesrdquoEconomicQuality Control vol 25 no 2 pp 153ndash167 2010
From Table 4 the experimenter could get the values of11988905
for different choices of 119888 and 119905119905005
in order to assert thatthe producerrsquos risk is less than 005 In this example the valueof11988905
should be 79120 for 119888 = 1 119905051199050
05= 20 and119901lowast = 075
This means that the product can have a life of 79120 timesthe required 50th percentile life time in order that under theabove acceptance sampling plan the product is accepted withprobability at least 075
42 Example 2 The second data refers to the data obtainedfrom Aarset [21] It represents the lifetimes of 50 devices inhours The data set was reported in hours as 01 02 1 1 1 11 2 3 6 7 11 12 18 18 18 18 18 21 32 36 40 45 46 47 5055 60 63 63 67 67 67 67 72 75 79 82 82 83 84 84 84 8585 85 85 85 86 and 86 On the similar lines of Example 1for this data the goodness of fit for 50 observations showedthat 119877 = 09175
Suppose that wewould like to establish the unknown 50thpercentile life time for the devices to be at least 10 hours andthe life test would be ended at 15 hrs which should have ledto the ratio of 119905
051199050
05= 15 Thus with 119888 = 5 119901lowast = 099
the experimenter may take from Table 1 the sample size 119899whichmust be at least 15Thus the sampling plan to the givendata is given by (119899 119888 1199051199050
05) = (15 5 15) Since there are 13
items with a failure time less than or equal to 15 hours inthe given sample of 50 observations the lot is accepted as the
result indicates that the 50th percentile lifetime 11990505
is at least10 hours with a confidence level of 119901lowast = 099
5 Summary and Conclusions
This paper provides the minimum sample size required todecide upon acceptingrejecting a lot based on its specified50th percentile (median) when the data follows half normaldistribution Assuming that the size of the given sample
6 Journal of Quality and Reliability Engineering
Table 4 Minimum ratio for the acceptability of a lot with producerrsquos risk 005
is minimum we can use the tables in this paper to pickup acceptance number 119888 and the upper bound in order toacceptreject the lot for a given probability of acceptance say119901lowast This paper also provides the sensitivity of the sampling
plans in terms of theOCWe advise the industrial practitionerand the experimenter to adopt this plan in order to save thecost and time of the experiment This plan can be furtherstudied for many other distributions and various quality andreliability characteristics as future research
Acknowledgments
The authors thank the editor and the reviewers for theirhelpful suggestions comments and encouragement whichhelped in improving the final version of the paper
References
[1] B Epstein ldquoTruncated life tests in the exponential caserdquo Annalsof Mathematical Statistics vol 25 pp 555ndash564 1954
[2] M Sobel and J A Tischendrof ldquoAcceptance samplingwith skewlife test objectiverdquo inProceedings of the 5thNational SyposiumonReliability and Qualilty Control pp 108ndash118 Philadelphia PaUSA 1959
[3] H P Goode and J H K Kao ldquoSampling plans based on theWeibull distributionrdquo in Proceedings of the 7th National Sypo-sium on Reliability and Qualilty Control pp 24ndash40 Philadel-phia Pa USA 1961
[4] S S Gupta and P A Groll ldquoGamma distribution in acceptancesampling based on life testsrdquo Journal of the American StatisticalAssociation vol 56 pp 942ndash970 1961
[5] S S Gupta ldquoLife test sampling plans for normal and lognormaldistributionrdquo Technometrics vol 4 pp 151ndash175 1962
[6] F W Fertig and N R Mann ldquoLife-test sampling plans for two-parameter Weibull populationsrdquo Technometrics vol 22 pp165ndash177 1980
[7] R R L Kantam and K Rosaiah ldquoHalf Logistic distribution inacceptance sampling based on life testsrdquo IAPQR Transactionsvol 23 no 2 pp 117ndash125 1998
[8] A Baklizi ldquoAcceptance sampling based on truncated life testsin the pareto distribution of the second kindrdquo Advances andApplications in Statistics vol 3 pp 33ndash48 2003
[9] C-J Wu and T-R Tsai ldquoAcceptance sampling plans forbirnbaum-saunders distribution under truncated life testsrdquoInternational Journal of Reliability Quality and Safety Engineer-ing vol 12 no 6 pp 507ndash519 2005
[10] K Rosaiah and R R L Kantam ldquoAcceptance sampling plansbased on inverse Rayleigh distributionrdquo Economic QualityControl vol 20 no 2 pp 277ndash286 2005
[11] K Rosaiah R R L Kantam and Ch Santosh Kumar ldquoReli-ability test plans for exponnetiated log-logistic distributionrdquoEconomic Quality Control vol 21 pp 165ndash175 2006
[12] T-R Tsai and S-J Wu ldquoAcceptance sampling based on trun-cated life tests for generalized Rayleigh distributionrdquo Journal ofApplied Statistics vol 33 no 6 pp 595ndash600 2006
[13] N Balakrishnan V Leiva and J Lopez ldquoAcceptance sam-pling plans from truncated life tests based on the generalizedBirnbaum-Saunders distributionrdquo Communications in Statis-tics Simulation and Computation vol 36 no 3 pp 643ndash6562007
[14] G Srinivasa Rao M E Ghitany and R R L KantamldquoReliability test plans for Marshall-Olkin extended exponentialdistributionrdquo Applied Mathematical Sciences vol 3 no 53-56pp 2745ndash2755 2009
[15] G Srinivasa Rao M E Ghitany and R R L Kantam ldquoAneconomic reliability test plan for Marshall-Olkin extendedexponential distributionrdquoAppliedMathematical Sciences vol 5no 1ndash4 pp 103ndash112 2011
[16] M Aslam C-H Jun and M Ahmad ldquoA group samplingplan based on truncated life test for gamma distributed itemsrdquoPakistan Journal of Statistics vol 25 no 3 pp 333ndash340 2009
[17] G Srinivasa Rao M E Ghitany and R R L KantamldquoReliability test plans for Marshall-Olkin extended exponentialdistributionrdquo Applied Mathematical Sciences vol 3 no 53-56pp 2745ndash2755 2009
[18] Y L Lio T-R Tsai and S-J Wu ldquoAcceptance sampling plansfrom truncated life tests based on the birnbaum-saunders distri-bution for percentilesrdquoCommunications in Statistics Simulationand Computation vol 39 no 1 pp 119ndash136 2010
[19] G Srinivasa Rao and R R L Kantam ldquoAcceptance samplingplans from truncated life tests based on log-logistic distributionfor percentilesrdquoEconomicQuality Control vol 25 no 2 pp 153ndash167 2010
Suppose that wewould like to establish the unknown 50thpercentile life time for the devices to be at least 10 hours andthe life test would be ended at 15 hrs which should have ledto the ratio of 119905
051199050
05= 15 Thus with 119888 = 5 119901lowast = 099
the experimenter may take from Table 1 the sample size 119899whichmust be at least 15Thus the sampling plan to the givendata is given by (119899 119888 1199051199050
05) = (15 5 15) Since there are 13
items with a failure time less than or equal to 15 hours inthe given sample of 50 observations the lot is accepted as the
result indicates that the 50th percentile lifetime 11990505
is at least10 hours with a confidence level of 119901lowast = 099
5 Summary and Conclusions
This paper provides the minimum sample size required todecide upon acceptingrejecting a lot based on its specified50th percentile (median) when the data follows half normaldistribution Assuming that the size of the given sample
6 Journal of Quality and Reliability Engineering
Table 4 Minimum ratio for the acceptability of a lot with producerrsquos risk 005
is minimum we can use the tables in this paper to pickup acceptance number 119888 and the upper bound in order toacceptreject the lot for a given probability of acceptance say119901lowast This paper also provides the sensitivity of the sampling
plans in terms of theOCWe advise the industrial practitionerand the experimenter to adopt this plan in order to save thecost and time of the experiment This plan can be furtherstudied for many other distributions and various quality andreliability characteristics as future research
Acknowledgments
The authors thank the editor and the reviewers for theirhelpful suggestions comments and encouragement whichhelped in improving the final version of the paper
References
[1] B Epstein ldquoTruncated life tests in the exponential caserdquo Annalsof Mathematical Statistics vol 25 pp 555ndash564 1954
[2] M Sobel and J A Tischendrof ldquoAcceptance samplingwith skewlife test objectiverdquo inProceedings of the 5thNational SyposiumonReliability and Qualilty Control pp 108ndash118 Philadelphia PaUSA 1959
[3] H P Goode and J H K Kao ldquoSampling plans based on theWeibull distributionrdquo in Proceedings of the 7th National Sypo-sium on Reliability and Qualilty Control pp 24ndash40 Philadel-phia Pa USA 1961
[4] S S Gupta and P A Groll ldquoGamma distribution in acceptancesampling based on life testsrdquo Journal of the American StatisticalAssociation vol 56 pp 942ndash970 1961
[5] S S Gupta ldquoLife test sampling plans for normal and lognormaldistributionrdquo Technometrics vol 4 pp 151ndash175 1962
[6] F W Fertig and N R Mann ldquoLife-test sampling plans for two-parameter Weibull populationsrdquo Technometrics vol 22 pp165ndash177 1980
[7] R R L Kantam and K Rosaiah ldquoHalf Logistic distribution inacceptance sampling based on life testsrdquo IAPQR Transactionsvol 23 no 2 pp 117ndash125 1998
[8] A Baklizi ldquoAcceptance sampling based on truncated life testsin the pareto distribution of the second kindrdquo Advances andApplications in Statistics vol 3 pp 33ndash48 2003
[9] C-J Wu and T-R Tsai ldquoAcceptance sampling plans forbirnbaum-saunders distribution under truncated life testsrdquoInternational Journal of Reliability Quality and Safety Engineer-ing vol 12 no 6 pp 507ndash519 2005
[10] K Rosaiah and R R L Kantam ldquoAcceptance sampling plansbased on inverse Rayleigh distributionrdquo Economic QualityControl vol 20 no 2 pp 277ndash286 2005
[11] K Rosaiah R R L Kantam and Ch Santosh Kumar ldquoReli-ability test plans for exponnetiated log-logistic distributionrdquoEconomic Quality Control vol 21 pp 165ndash175 2006
[12] T-R Tsai and S-J Wu ldquoAcceptance sampling based on trun-cated life tests for generalized Rayleigh distributionrdquo Journal ofApplied Statistics vol 33 no 6 pp 595ndash600 2006
[13] N Balakrishnan V Leiva and J Lopez ldquoAcceptance sam-pling plans from truncated life tests based on the generalizedBirnbaum-Saunders distributionrdquo Communications in Statis-tics Simulation and Computation vol 36 no 3 pp 643ndash6562007
[14] G Srinivasa Rao M E Ghitany and R R L KantamldquoReliability test plans for Marshall-Olkin extended exponentialdistributionrdquo Applied Mathematical Sciences vol 3 no 53-56pp 2745ndash2755 2009
[15] G Srinivasa Rao M E Ghitany and R R L Kantam ldquoAneconomic reliability test plan for Marshall-Olkin extendedexponential distributionrdquoAppliedMathematical Sciences vol 5no 1ndash4 pp 103ndash112 2011
[16] M Aslam C-H Jun and M Ahmad ldquoA group samplingplan based on truncated life test for gamma distributed itemsrdquoPakistan Journal of Statistics vol 25 no 3 pp 333ndash340 2009
[17] G Srinivasa Rao M E Ghitany and R R L KantamldquoReliability test plans for Marshall-Olkin extended exponentialdistributionrdquo Applied Mathematical Sciences vol 3 no 53-56pp 2745ndash2755 2009
[18] Y L Lio T-R Tsai and S-J Wu ldquoAcceptance sampling plansfrom truncated life tests based on the birnbaum-saunders distri-bution for percentilesrdquoCommunications in Statistics Simulationand Computation vol 39 no 1 pp 119ndash136 2010
[19] G Srinivasa Rao and R R L Kantam ldquoAcceptance samplingplans from truncated life tests based on log-logistic distributionfor percentilesrdquoEconomicQuality Control vol 25 no 2 pp 153ndash167 2010
is minimum we can use the tables in this paper to pickup acceptance number 119888 and the upper bound in order toacceptreject the lot for a given probability of acceptance say119901lowast This paper also provides the sensitivity of the sampling
plans in terms of theOCWe advise the industrial practitionerand the experimenter to adopt this plan in order to save thecost and time of the experiment This plan can be furtherstudied for many other distributions and various quality andreliability characteristics as future research
Acknowledgments
The authors thank the editor and the reviewers for theirhelpful suggestions comments and encouragement whichhelped in improving the final version of the paper
References
[1] B Epstein ldquoTruncated life tests in the exponential caserdquo Annalsof Mathematical Statistics vol 25 pp 555ndash564 1954
[2] M Sobel and J A Tischendrof ldquoAcceptance samplingwith skewlife test objectiverdquo inProceedings of the 5thNational SyposiumonReliability and Qualilty Control pp 108ndash118 Philadelphia PaUSA 1959
[3] H P Goode and J H K Kao ldquoSampling plans based on theWeibull distributionrdquo in Proceedings of the 7th National Sypo-sium on Reliability and Qualilty Control pp 24ndash40 Philadel-phia Pa USA 1961
[4] S S Gupta and P A Groll ldquoGamma distribution in acceptancesampling based on life testsrdquo Journal of the American StatisticalAssociation vol 56 pp 942ndash970 1961
[5] S S Gupta ldquoLife test sampling plans for normal and lognormaldistributionrdquo Technometrics vol 4 pp 151ndash175 1962
[6] F W Fertig and N R Mann ldquoLife-test sampling plans for two-parameter Weibull populationsrdquo Technometrics vol 22 pp165ndash177 1980
[7] R R L Kantam and K Rosaiah ldquoHalf Logistic distribution inacceptance sampling based on life testsrdquo IAPQR Transactionsvol 23 no 2 pp 117ndash125 1998
[8] A Baklizi ldquoAcceptance sampling based on truncated life testsin the pareto distribution of the second kindrdquo Advances andApplications in Statistics vol 3 pp 33ndash48 2003
[9] C-J Wu and T-R Tsai ldquoAcceptance sampling plans forbirnbaum-saunders distribution under truncated life testsrdquoInternational Journal of Reliability Quality and Safety Engineer-ing vol 12 no 6 pp 507ndash519 2005
[10] K Rosaiah and R R L Kantam ldquoAcceptance sampling plansbased on inverse Rayleigh distributionrdquo Economic QualityControl vol 20 no 2 pp 277ndash286 2005
[11] K Rosaiah R R L Kantam and Ch Santosh Kumar ldquoReli-ability test plans for exponnetiated log-logistic distributionrdquoEconomic Quality Control vol 21 pp 165ndash175 2006
[12] T-R Tsai and S-J Wu ldquoAcceptance sampling based on trun-cated life tests for generalized Rayleigh distributionrdquo Journal ofApplied Statistics vol 33 no 6 pp 595ndash600 2006
[13] N Balakrishnan V Leiva and J Lopez ldquoAcceptance sam-pling plans from truncated life tests based on the generalizedBirnbaum-Saunders distributionrdquo Communications in Statis-tics Simulation and Computation vol 36 no 3 pp 643ndash6562007
[14] G Srinivasa Rao M E Ghitany and R R L KantamldquoReliability test plans for Marshall-Olkin extended exponentialdistributionrdquo Applied Mathematical Sciences vol 3 no 53-56pp 2745ndash2755 2009
[15] G Srinivasa Rao M E Ghitany and R R L Kantam ldquoAneconomic reliability test plan for Marshall-Olkin extendedexponential distributionrdquoAppliedMathematical Sciences vol 5no 1ndash4 pp 103ndash112 2011
[16] M Aslam C-H Jun and M Ahmad ldquoA group samplingplan based on truncated life test for gamma distributed itemsrdquoPakistan Journal of Statistics vol 25 no 3 pp 333ndash340 2009
[17] G Srinivasa Rao M E Ghitany and R R L KantamldquoReliability test plans for Marshall-Olkin extended exponentialdistributionrdquo Applied Mathematical Sciences vol 3 no 53-56pp 2745ndash2755 2009
[18] Y L Lio T-R Tsai and S-J Wu ldquoAcceptance sampling plansfrom truncated life tests based on the birnbaum-saunders distri-bution for percentilesrdquoCommunications in Statistics Simulationand Computation vol 39 no 1 pp 119ndash136 2010
[19] G Srinivasa Rao and R R L Kantam ldquoAcceptance samplingplans from truncated life tests based on log-logistic distributionfor percentilesrdquoEconomicQuality Control vol 25 no 2 pp 153ndash167 2010
is minimum we can use the tables in this paper to pickup acceptance number 119888 and the upper bound in order toacceptreject the lot for a given probability of acceptance say119901lowast This paper also provides the sensitivity of the sampling
plans in terms of theOCWe advise the industrial practitionerand the experimenter to adopt this plan in order to save thecost and time of the experiment This plan can be furtherstudied for many other distributions and various quality andreliability characteristics as future research
Acknowledgments
The authors thank the editor and the reviewers for theirhelpful suggestions comments and encouragement whichhelped in improving the final version of the paper
References
[1] B Epstein ldquoTruncated life tests in the exponential caserdquo Annalsof Mathematical Statistics vol 25 pp 555ndash564 1954
[2] M Sobel and J A Tischendrof ldquoAcceptance samplingwith skewlife test objectiverdquo inProceedings of the 5thNational SyposiumonReliability and Qualilty Control pp 108ndash118 Philadelphia PaUSA 1959
[3] H P Goode and J H K Kao ldquoSampling plans based on theWeibull distributionrdquo in Proceedings of the 7th National Sypo-sium on Reliability and Qualilty Control pp 24ndash40 Philadel-phia Pa USA 1961
[4] S S Gupta and P A Groll ldquoGamma distribution in acceptancesampling based on life testsrdquo Journal of the American StatisticalAssociation vol 56 pp 942ndash970 1961
[5] S S Gupta ldquoLife test sampling plans for normal and lognormaldistributionrdquo Technometrics vol 4 pp 151ndash175 1962
[6] F W Fertig and N R Mann ldquoLife-test sampling plans for two-parameter Weibull populationsrdquo Technometrics vol 22 pp165ndash177 1980
[7] R R L Kantam and K Rosaiah ldquoHalf Logistic distribution inacceptance sampling based on life testsrdquo IAPQR Transactionsvol 23 no 2 pp 117ndash125 1998
[8] A Baklizi ldquoAcceptance sampling based on truncated life testsin the pareto distribution of the second kindrdquo Advances andApplications in Statistics vol 3 pp 33ndash48 2003
[9] C-J Wu and T-R Tsai ldquoAcceptance sampling plans forbirnbaum-saunders distribution under truncated life testsrdquoInternational Journal of Reliability Quality and Safety Engineer-ing vol 12 no 6 pp 507ndash519 2005
[10] K Rosaiah and R R L Kantam ldquoAcceptance sampling plansbased on inverse Rayleigh distributionrdquo Economic QualityControl vol 20 no 2 pp 277ndash286 2005
[11] K Rosaiah R R L Kantam and Ch Santosh Kumar ldquoReli-ability test plans for exponnetiated log-logistic distributionrdquoEconomic Quality Control vol 21 pp 165ndash175 2006
[12] T-R Tsai and S-J Wu ldquoAcceptance sampling based on trun-cated life tests for generalized Rayleigh distributionrdquo Journal ofApplied Statistics vol 33 no 6 pp 595ndash600 2006
[13] N Balakrishnan V Leiva and J Lopez ldquoAcceptance sam-pling plans from truncated life tests based on the generalizedBirnbaum-Saunders distributionrdquo Communications in Statis-tics Simulation and Computation vol 36 no 3 pp 643ndash6562007
[14] G Srinivasa Rao M E Ghitany and R R L KantamldquoReliability test plans for Marshall-Olkin extended exponentialdistributionrdquo Applied Mathematical Sciences vol 3 no 53-56pp 2745ndash2755 2009
[15] G Srinivasa Rao M E Ghitany and R R L Kantam ldquoAneconomic reliability test plan for Marshall-Olkin extendedexponential distributionrdquoAppliedMathematical Sciences vol 5no 1ndash4 pp 103ndash112 2011
[16] M Aslam C-H Jun and M Ahmad ldquoA group samplingplan based on truncated life test for gamma distributed itemsrdquoPakistan Journal of Statistics vol 25 no 3 pp 333ndash340 2009
[17] G Srinivasa Rao M E Ghitany and R R L KantamldquoReliability test plans for Marshall-Olkin extended exponentialdistributionrdquo Applied Mathematical Sciences vol 3 no 53-56pp 2745ndash2755 2009
[18] Y L Lio T-R Tsai and S-J Wu ldquoAcceptance sampling plansfrom truncated life tests based on the birnbaum-saunders distri-bution for percentilesrdquoCommunications in Statistics Simulationand Computation vol 39 no 1 pp 119ndash136 2010
[19] G Srinivasa Rao and R R L Kantam ldquoAcceptance samplingplans from truncated life tests based on log-logistic distributionfor percentilesrdquoEconomicQuality Control vol 25 no 2 pp 153ndash167 2010